Embedding Long Paths in k-Ary n-Cubes with Faulty Nodes and Links

Let k ges 4 be even and let n ges 2. Consider a faulty k-ary n-cube Q
^k

_n
in which the number of node faults fv and the number of link faults f
_e
are such that f
_v
+ f
_e
les 2n-2. We prove that given any two healthy nodes s and e of Q
^k

_n
, there is a path from s to e of length at least k
ⁿ
- 2 f
_v
-1 (respectively, k
ⁿ
- 2 f
_v
-2) if the nodes s and e have different (respectively the same) parities (the parity of a node in Q
^k

_n
is the sum modulo 2 of the elements in the n-tuple over {0,1,...,k 1} representing the node). Our result is optimal in the sense that there are pairs of nodes and fault configurations for which these bounds cannot be improved, and it answers questions recently posed by Yang, Tan et al. (2007) and by Fu (2006). Furthermore, we extend known results, obtained by Kim and Park (2000), for the case when n=2.